polyhedron

Combinatorics

In combinatorics a polyhedron is the solution set of a finite system
of linear inequalities. The solution set is in ℝn for integer
n. Hence, it is a convex set. Each extreme point of such a polyhedron is also called a vertex (or corner point) of the polyhedron. A solution
set could be empty. If the solution set is bounded (that is, is contained in
some sphere) the polyhedron is said to be bounded.

Elementary Geometry

In elementarygeometry a polyhedron is a solid bounded by a finite number of plane faces,
each of which is a polygon. This of course is not a precise definition as it
relies on the undefined term “solid”. Also, this definition allows a polyhedron
to be non-convex.

Careful Treatments of Geometry

In treatments of geometry that are carefully done a definition due to Lennes is
sometimes used [2]. The intent is to rule out certain objects that one does not want
to consider and to simplify the theory of dissection.
A polyhedron is a set of points consisting of a finite set of
trianglesT, not all coplanar, and their interiors such that

(i)

every side of a triangle is common to an even number of triangles of the
set, and

(ii)

there is no subset T′ of T such that (i) is true of a proper subset
of T′.

Notice that condition (ii) excludes, for example, two cubes that are disjoint. But two
tetrahedra having a common edge are allowed. The faces of the polyhedron are the insides
of the triangles. Note that the condition that the faces be triangles
is not important, since a polygon an be dissected into triangles.
Also note since a triangle meets an even number of other triangles,
it is possible to meet 4,6 or any other even number of triangles. So for example,
a configuration of 6 tetrahedra all sharing a common edge is allowed.

By dissections of the triangles one can create a set of triangles in which
no face intersects another face, edge or vertex. If this done the
polyhedron is said to be .

A convex polyhedron is one such that all its inside points lie on one side of
each of the planes of its faces.

An Euler polyhedronP is a set of points consisting of a finite set
of polygons, not all coplanar, and their insides such that

(i)

each edge is common to just two polygons,

(ii)

there is a way using edges of P from a given vertex to each vertex, and

(iii)

any simple polygonp made up of edges of P, divides the polygons
of P into two sets A and B such that any way, whose points are on P
from any point inside a polygon of A to a point inside a polygon of B,
meets p.

It is a theorem, proved here (http://planetmath.org/ClassificationOfPlatonicSolids), that for a regular polyhedron, the number of polygons with the same
vertex is the same for each vertex and that there are 5 types of regular polyhedra.

Notice that a cone, and a cylinder are not polyhedra since they have “faces” that are not polygons.

A simple polyhedron is one that is homeomorphic to a sphere. For such a polyhedron
one has V-E+F=2, where V is the number of vertices, E is the number of edges
and F is the number of faces. This is called Euler’s formula.